Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ The owner of a new restaurant is designing the floor plan, and he is deciding between two different seating arrangements. The first plan consists of $18$ tables and $20$ booths, which will seat a total of $290$ people. The second plan consists of $20$ tables and $25$ booths, which will seat a total of $350$ people. How many people can be seated at each type of table? $\newline$ Every table can seat $\_$ people, and every booth can seat $\_$ people.

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Solve a system of equations using elimination: word problems

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.

\newline

The owner of a new restaurant is designing the floor plan, and he is deciding between two different seating arrangements. The first plan consists of

18

tables and

20

booths, which will seat a total of

290

people. The second plan consists of

20

tables and

25

booths, which will seat a total of

350

people. How many people can be seated at each type of table?

\newline

Every table can seat

\_

people, and every booth can seat

\_

people.

Define Variables: Let's denote the number of people that can be seated at a table as $t$ and the number of people that can be seated at a booth as $b$ . The first plan consists of $18$ tables and $20$ booths seating a total of $290$ people, which gives us the equation $18t + 20b = 290$ .
First Plan: The second plan consists of $20$ tables and $25$ booths seating a total of $350$ people, which gives us the equation $20t + 25b = 350$ .
Second Plan: We now have a system of two equations. We need to eliminate one of the variables, $t$ or $b$ . We choose to eliminate $t$ because its coefficients are close to each other, which might make the calculations simpler.
Eliminate Variable: To eliminate $t$ , we can multiply the first equation by $20$ and the second equation by $18$ , so the coefficients of $t$ in both equations are the same. This gives us the new equations $360t + 400b = 5800$ (first equation multiplied by $20$ ) and $360t + 450b = 6300$ (second equation multiplied by $18$ ).
Subtract Equations: We now subtract the first new equation from the second new equation to eliminate $t$ . This gives us $50b = 500$ .
Find Booth Capacity: Solving for $b$ , we divide both sides of the equation by $50$ , which gives us $b = 10$ . This means that every booth can seat $10$ people.
Substitute in Equation: We substitute $b = 10$ into the first original equation $18t + 20b = 290$ . This gives us $18t + 20(10) = 290$ .
Solve for Tables: Solving for $t$ , we first calculate $20(10)$ which is $200$ , and then we subtract $200$ from $290$ , which gives us $18t = 90$ .
Final Solution: Finally, we divide both sides of the equation by $18$ to find $t$ , which gives us $t = 5$ . This means that every table can seat $5$ people.